An (m, M) machine repair problem with spares and state dependent rates: A diffusion process approach

The diffusion approximation model for (m, M) general machine repair problems with spare part support has been developed. The inter-failure time and repair times are assumed to be generally identically and independently distributed (i.i.d.). The failure rate of operating units in a short system when fewer than M units are operating and all spares are being used, is faster than a normal system. The spare units may also fail with rates different from operating units. The repairmen switch to the faster rate to reduce a backlog of down units in the case of a busy repair facility. By using reflecting boundaries, the approximate formulae for some performance measures, namely, expected number of inoperative/operative units and probability that the system is short/down have been obtained.     

Logistics Availability for Sites with Periodic or Random Resupply

Logistics Availability (AL), the probability that a system is not in a spares delay downstate at any instant of time, is a basic element of Operational Availability. Formulas for AL for serial systems of assemblies are derived. Each assembly has exponential times to failure and is supported by a spares inventory that is either periodically resupplied or resupplied as-needed (an order is placed following each failure which requires a spare to restore the system). The formulas depend upon the reliability of the operating assemblies, the specified number of spares in the full-up inventory, the usual resupply times, and the constant spares delay downtime for failures that occur when there are no spares in the inventory. These formulas for AL can be used in well-known integer optimizing processes to select the spares quantities for the site inventory and/or to determine acceptable resupply periods and spares delay downtime, in addition to assessing AL.     

Cost analysis of a two-unit standby system with delayed replacement and better utilization of units

This paper develops the model for a system, having two identical units—one operative and the other cold standby. Each unit of the system has three modes—normal, partial failure and total failure. The replacement time of a failed unit by a standby unit is not negligible but is a random variable. System fails when both the units fail totally. Failure time distributions of units are exponential, whereas repair time distributions are arbitrary. Several reliability characteristics of interest to system designers and operations managers have been evaluated using the theory of regeneration point technique.     

A rapid smart approach for reliability and failure mode analysis of memories and large systems

This paper presents a novel system simulation methodology based on the known Monte Carlo technique, used for reliability and failure mode analysis of complex and large systems. The presented approach, called “state-merging and assorted random-testing” (SMART), is particularly applicable to systems involving different types of clusters of identical components, and is ideally suited for simulation of huge memories and similar systems. Simulators based on this approach are insensitive to the number of system components, system reliability or the number of associated spares or standby units, and thus they afford an extremely small simulation time compared to the accelerated Monte Carlo simulation time.     

Cost analysis of a two-unit chargeable standby system with interchangeable units and two types of failure

This paper studies the analysis of a stochastic model related to a two-unit chargeable standby system with interchangeable units (identical), i.e. the operative and standby units are interchanged at random epochs. The system can fail either due to power fluctuations or due to the operator's inefficiency. Failure time distributions are negative exponential while the distributions of repair times and time to interchange (of units) are arbitrary. Using a regenerative point technique, we have obtained various reliability characteristics to carry out the cost-benefit analysis.     

Reliability of a system with warm standbys and repairmen

This paper studies the reliability characteristics of a system with M operating machines, S warm standby spares and R repairmen in the repair facility. Failure times and repair times are assumed to be exponentially distributed. The k out of M+S system is analyzed where k = 1, 2, …, M, and numerical results are provided.     

Stochastic analysis of a multi-unit cold standby system working in orbit form

This paper considers the analysis of a single server n-similar unit system. Initially k(<n) units form an orbit which functions if one unit functions at a time and the remaining n − k units work as cold standbys. When a unit fails in the orbit it is instantaneously replaced by one of the standbys with the help of a perfect transfer switch. The system is said to fail when n − k + 1 units have failed. The distribution of time to failure and time to repair of a unit are negative exponential. Using the regenerative point technique several reliability characteristics are obtained to carry out the cost-benefit analysis.     

Common-cause failures and critical human errors in repairable and non-repairable systems

This paper presents a reliability analysis of a k-out-of-N:G redundant system with common-cause failures, critical human errors and r repair facilities. The system is in a failed state when common-cause failures or critical human errors occurred or k units failed. When less than k units failed, the failed units are to be repaired. If the whole system is in a failed state, it cannot be repaired. Laplace transorms of state probabilities and system reliability are derived. Various versions of mean time to failure of a system are also reported.     

A k-out-of-N:G redundant system with cold standby units and common-cause failures

This paper presents a k-out-of N:G redundant system with M cold standby units, r repair facilities and common-cause failures. The constant failure rates of the operating and cold standby units are different. Failed system repair times are arbitrarily distributed. The system is in a failed state when (N+M?k+1) units failed or a common-cause occurred. Laplace transforms of the state probabilities, the availability of the system and the system steady-state availability are derived.     

Analysis of a single unit system with helping unit

This paper considers a single unit (hence forth we call it a main unit) system with a helping unit. Initially both units are operative. The functioning of the main unit is assumed to be dependent on the helping unit, if the helping unit is in the operative mode, otherwise the main unit works independently with an increasing failure rate. The main unit can fail either partially or totally, while the helping unit can fail only totally. Failure times are assumed to follow negative exponential distributions, while the repair-time distributions are general. Important characteristics related to the system are obtained by using the regeneration point technique.     

Availability calculations with exhaustible spares

Many systems rely solely on the spares they carry to fulfil their missions. The authors develop relatively simple equations for the availability of a system with exhaustible spares. The equations are conservative but, for a large system, are more tractable than simulation or the exact approach based on Markov theory. The equations are useful for tradeoff or sensitivity analysis. Given various complements of spares, system availability can be calculated, or the optimal selection of spares can be determined. Since the equations conservatively approximate system availability if the system consists of several types of units in series, the equations can be used to determine if a system meets its availability requirement. If the calculated system availability is less than the requirement, spares could be added or more exact techniques could be applied. Because of various simplifying assumptions, the equations are most exact when repair time is small compared to mission time and when k is close to n for k-out-of-n:G systems     

A Reliability and Comparative Analysis of Two Standby System Configurations

Equations are derived which enable one to calculate the system reliability for parallel or triple modular redundant systems with standby spares. Software error detection is introduced into the TMR/Spares system configuration in order to utilize fully all of the units. An indication of the sensitivity of the system reliability to an increase in the number of spares, partitioning, switching, variations in the powered and unpowered failures rates, and time is presented. A comparison of the parallel and the TMR/Spares system configurations, under similar conditions, is given.     

On the transient analysis of a maintained system subject to random stresses

This paper deals with the probablistic analysis of a system subject to stresses. The system consists of a basic unit and a standby and is provided with a service facility to carry out maintenance, inspection prior to repair and repair of the units in the system. The operating unit is sent for maintenance as soon as its strength, after being hit by a stress, falls below a specified critical value, the strength being assumed to be deterministic. The operating unit may also fail on any stress by virtue of the stress exceeding the strength. The failed unit is subject to inspection prior to repair to ascertain the type of repair the unit has to undergo. The stress experienced by a unit is assumed to be a random variable governed by a probability law. The time between successive stresses and the time for maintenance, inspection and repair are random variables governed by probability laws. Explicit expressions for various system characteristics have been obtained using the state-space method and the regeneration point technique.     

Reliability Analysis of the k-out-of-n:F System

A class of repairable systems known as k-out-of-n:F systems, 1 ? k ? n, consists of n units in parallel redundancy which are serviced by a single repairman; system failure occurs when k units are simultaneously inoperable for the first time. In this paper, assuming constant failure rates and general repair distributions, reliability characteristics of the k-out-of-n:F system are treated using two different methods. In Part I, a conditional transform approach is applied to the 2-out-of-n:F system. Transforms of distributions are obtained for T (the time to system failure), the time spent on repairs during (0, T) and the free time of the repairman during (0, T). In Part II, the supplementary variable technique is used to investigate time to failure characteristics of the k-out-of-n:F system for k = 2 and k = 3. A model of an airport limousine service illustrates the use of the results.     

Reliability of Special Redundant Systems Considering Exchange Time and Repair Time

The reliability of a 2-out-of-3 parallel redundant system having a limited number of standby spare units is derived when the exchange of the failed unit for a spare unit is not instantaneous. The reliability can be represented in the form of a failure state diagram. When the number of standby spares and the repair rate are both small, the influence of the exchange rate is small. When the number of standby spares and the repair rate are both large, the influence of the exchange rate is large. The number of standby spares and the repair rate influence the probability that system failure occurs after all spare units have failed. The exchange rate strongly influences the probability of system failure during the exchange time.     

Optimum Age Replacement in the Bivariate Exponential Case

An age replacement policy is considered for pairs of units which operate in parallel and which have lifetimes displaying a bivariate exponential distribution. Both units are to be replaced at the same time. The limiting expected cost per unit time is the optimization criterion. The results state that no replacements should be made until at least one of the units in the pair fails. Both units shoould then be replaced either when one fails or when both fail, depending on which procedure involves the smaller limiting expected cost per unit time.     

Using system reliability to determine supportability turnaroundtime at a repair depot

This paper uses an expression for system reliability at a repair depot to construct a nonlinear, nonpolynomial function which is amenable to numerical analysis and has a zero equal to the supportability turnaround time (STAT) for a failed unit. System reliability is in terms of the constant failure rate for all units, number of spares on-hand at the time a unit fails, and projected repair completion dates for up to four unrepaired units. In this context, STAT represents the longest repair time (for a failed unit) which assures a given reliability level; system reliability is the probability that spares are ready to replace failed units during the STAT period. The ability to calculate STAT-values is important for two reasons: (1) subtraction of the repair time for a failed unit from its STAT-value yields the latest repair start-time (for this unit) which assures a desired reliability, and (2) the earlier the latest repair start-time, the higher the priority for starting the repair of this unit. Theorems show the location of STAT with respect to the list of repair completion dates, and form the foundation of the root-finding-based algorithm for computing STAT-values. Numerical examples illustrate the algorithm     

Cost analysis of a two-unit standby system with two types of repairmen

This paper is concerned with a two-unit cold standby system with two types of repairmen. One “regular” repairman is kept for repairing the units as soon as they fail. It is assumed that sometimes he might not be able to do the repairs within some tolerable time (patience time). Another “expert” repairman, assumed to be perfect, is called on to do the repairs on the completion of this patience time or on the failure of the system, whichever is later.Various measures of system effectiveness are calculated using semi-Markov processes and regenerative processes. Based on these measures, a rule is developed whether the expert repairman should be called after the system failure. Further numerical results for a case, in which repair time and patience time both have non-Markovian property, are also investigated. Then the upper bound of the cost K₃, below which the expert repairman should be called immediately after the system failure and the corresponding increase in profit are calculated.     

Parametric Programming Approach for a Two-Unit Repairable System With Imperfect Coverage, Reboot and Fuzzy Parameters

This paper proposes a procedure for constructing the membership functions of the system characteristics of a two-unit repairable system in which the coverage factor for an operating unit failure is possibly considered. Times to failure, and times to repair of the operating units are assumed to follow fuzzified exponential distributions. In addition, the recovery times, and reboot times of the failed units also follow fuzzified exponential distributions. The $alpha$-cut approach is used to extract the fuzzy repairable system from a family of conventional crisp intervals for the desired system characteristics, determined with a set of parametric nonlinear programs using their membership functions. Two numerical examples illustrate the practicality of the proposed approach. Because the system characteristics are governed by the membership functions, more information is provided for use by designers and practitioners. The successful extension of the parameter spaces to fuzzy environments permits the repairable system to have wider practical applications.      

Stochastic analysis of systems with primary and secondary failures

This paper presents the stochastic analysis of repairable systems involving primary as well as secondary failures. To this end, two models are considered. The first model represents a system with two identical warm standbys. The failure rates of units and the system are constant and independent while the repair times are arbitrarily distributed. The second system modeled consists of three repairable regions. The system operates normally if all three regions are operating, otherwise it operates at a derated level unless all three regions fail. The failure rates and repair times of the regions are constant and independent. The first model is analyzed using the supplemental variable technique while the second model is analyzed using the regenerative point technique in the Markov renewal process. Various expressions including system availability, system reliability and mean time to system failure are obtained.